**Background**

The local density approximation (LDA), the earliest approximation in DFT, constructs the local exchange-correlation energy density at a point in space from just the local electron density there. Derived from and exact for any uniform electron gas, LDA tends to minimize the inhomogeneity of electron densities of real materials and overestimates the strengths of all bonds near equilibrium. By building in the electron density gradient to reduce this tendency, generalized gradient approximations (GGAs) soften the bonds. Depending on how the electron density gradient is built in, a GGA can be rather accurate for structures or energies, but not both. This dilemma reflects a formal limitation: A GGA cannot satisfy all the known exact constraints appropriate to a semilocal functional (LDA, GGAs, and meta-GGAs) where the exchange-correlation energy is efficiently evaluated as a single integral over three-dimensional space. By mixing GGAs with nonlocal exact exchange, hybrid GGAs can further improve the description of covalent, ionic, and hydrogen bonds. However, hybrid GGAs still fail to describe vdW interactions. The computational cost of a hybrid functional can be 10 to 100 times that of a semilocal functional in standard plane-wave codes, even more so for metallic systems. Another problem with hybrids is that a universal exact-exchange mixing parameter is not determined by any exact condition.

The inclusion of the kinetic energy density in addition to the electron density and its gradient enables meta-GGAs to have the flexibility to satisfy more exact constraints and thus circumvent the “structure or energy” dilemma experienced by GGAs. By using a dimensionless variable alpha, meta-GGAs can recognize the slowly-varying densities (alpha~1, characterizing metallic bonds), the single-orbital systems (alpha=0, characterizing covalent single bonds), and non-covalent bonds with between two closed shells (alpha>>1). Alpha is directly related to the electron localization function (ELF) and therefore identifies different chemical bonds.

The inclusion of the kinetic energy density in addition to the electron density and its gradient enables meta-GGAs to have the flexibility to satisfy more exact constraints and thus circumvent the “structure or energy” dilemma experienced by GGAs. By using a dimensionless variable alpha, meta-GGAs can recognize the slowly-varying densities (alpha~1, characterizing metallic bonds), the single-orbital systems (alpha=0, characterizing covalent single bonds), and non-covalent bonds with between two closed shells (alpha>>1). Alpha is directly related to the electron localization function (ELF) and therefore identifies different chemical bonds.